For example, look at the book “Our crowd” written by a secular Jew totally uninterested in halacha. The book has a totally unrelated purpose, And then check how many of the families mentioned (who came to the United States before the American Civil War) were intermarried. Furthermore, due to the American custom of women adopting the husband’s surname, it means that the people with “Jewish” names are the least likely to be Jewish.
Among Eastern European Jews, the movement towards assimilation began only in the late 19th century, so it is easier to trace.
How many “Reform” Jews are motivated by ideology is questionable. In a typical Reform “Temple” there is a small inner-circle of ideological “Reform”, and a large number of “twice-a-year” types whose motivation is a desire to have an unburdened Jewish affiliation. If you do not county the nominal Reform as part of Reform, i.e. base the estimate of who is a non-assimilated Jew on who engages in some form of Jewish religious life on a daily basis, then instead of Reform and Conservative dominating American Jewry (the conventional analysis based on synagogue memebers and “high holiday” attendance) you end up with Orthodox Jews (all of whom engage in “Jewish” activities daily even if only making a bracha or wearing distinctive clothes) as the overwhelming majority of American Jews.
Most non-Orthodox Jews in America perceive themselves as Americans and have little interest or knowledge of Yiddishkeit. From all perspectives, political, social, ecnomical and increasingly genetic, they are a distinct people from the frum community. The split is largely complete. Asking what to do about assimilation today makes no more sense that asking what to do about the holocaust — it’s done, it’s over, its history, and we need to live with the consequences.